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Hausdorff Dimensions of Self-Similar and Self-Affine Fractals in the Heisenberg Group |
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| Authors: | Balogh, Zoltan M. Tyson, Jeremy T. |
| Affiliation: | Department of Mathematics, University of Bern Sidlerstrasse 5, 3012 Bern, Switzerland. E-mail: zoltan.balogh{at}math-stat.unibe.ch Department of Mathematics, University of Illinois 1409 W. Green Street, Urbana, IL 61801, USA. E-mail: tyson{at}math.uiuc.edu |
| Abstract: | We study the Hausdorff dimensions of invariant sets for self-similarand self-affine iterated function systems in the Heisenberggroup. In our principal result we obtain almost sure formulaefor the dimensions of self-affine invariant sets, extendingto the Heisenberg setting some results of Falconer and Solomyakin Euclidean space. As an application, we complete the proofof the comparison theorem for Euclidean and Heisenberg Hausdorffdimension initiated by Balogh, Rickly and Serra-Cassano. 2000Mathematics Subject Classification 22E30, 28A78 (primary), 26A18,28A78 (secondary). |
| Keywords: | Heisenberg group Hausdorff dimension iterated function system |
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